Lecture 3

(Recap of Part 2) Dirac spinor 2 component Weyl spinors Raising and lowering of indices

(Recap of Part 2) Dirac spinor 2 component Weyl spinors

Anti-commuting “c-numbers” {complex numbers } If{Grassmann numbers} then Superspace (Recap of Part 2) Grassmann Numbers Lorentz transformations act on Minkowski space-time: In supersymmetric extensions of Minkowki space-time, SUSY transformations act on a superspace: 8 coordinates, 4 space time, 4 fermionic Grassmann numbers

Part 3

SUSY transformation Recall Poincare transformation from lecture 1: Similarly for SUSY: But Baker-Campbell-Hausdorff formula applies here! Home exercise check:

SUSY transformation Independent of x, so global SUSY transformation Excercise: for the enthusiastic check these satisfy the SUSY algebra given earlier

General Superfield (where we have suppressed spinor indices) Scalar fieldspinor Scalar field Vector field spinor Scalar field Total derivative SUSY transformation should give a function of the same form, ) component fields transformations

General Superfield (where we have suppressed spinor indices) Scalar fieldspinor Scalar field Vector field spinor Scalar field Total derivative SUSY transformation should give a function of the same form, ) component fields transformations Invariant SUSY contribution to action

2.3 General Superfield (where we have suppressed spinor indices) Scalar fieldspinor Scalar field Vector field spinor Scalar field SUSY transformation should give a function of the same form, ) component fields transformations Total derivative

Notes: 1.) ± D is four divergence ) any such “D-term” in a Lagrangian will yield an action invariant under supersymmetric transformations 2.) Linear combinations and products of superfields are also superfields, e.g. is a superfields if are superfields. 3.) This is the general superfield, but it does not form an Irreducible representation of SUSY. 4.) Irreducible representations of supersymmetry, chiral superfields and vector superfields will now be discussed.

Chiral Superfields - Irreducible multiplet, - Describes lepton / slepton, quark / squark and Higgs / Higgsino multiplets Try: But

Chiral Superfields - Irreducible multiplet, - Describes lepton / slepton, quark / squark and Higgs / Higgsino multiplets Try: But

Chiral Superfields - Irreducible multiplet, - Describes lepton / slepton, quark / squark and Higgs / Higgsino multiplets Try: But

scalar spinor For example?? Auxilliary field (explained soon) Note: 2 complex scalars ) 4 d.o.f. 1 complex spinor ) 4 d.o.f. Different representations of the SUSY algebra Home exercise

Working in the “ Chiral” representation, the SUSY transformation of a left chiral superfield is given by, Chiral representation of SUSY generators Four-divergence, yields invariant action under SUSY F-terms provide contributions to the Lagrangian density Boson ! fermion Fermion ! boson